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Summer Pridgen [tex]$\quad 07 / 16 / 24 \, 6:59 \, \text{PM}$[/tex]

Question [tex]$18, \, 3.2.59$[/tex]
HW Score: [tex]$61.73\%, \, 16.67$[/tex] of 27 points
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For the following, find the discriminant, [tex]$b^2 - 4ac$[/tex], and then determine whether one real-number solution, two different real-number solutions, or two different imaginary number solutions exist.

[tex]$x^2 + 2x + 3 = 0$[/tex]

What is the discriminant, [tex]$b^2 - 4ac$[/tex]?
[tex]$\boxed{\text{Simplify your answer.}}$[/tex]



Answer :

GinnyAnswer
Let's solve the problem step-by-step:

1. Identify the coefficients of the quadratic equation:
[tex]\[ x^2 + 2x + 3 = 0 \][/tex]
Here, the coefficients are:
[tex]\[ a = 1, \quad b = 2, \quad c = 3 \][/tex]

2. Calculate the discriminant [tex]\(\Delta\)[/tex] using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[ \Delta = 2^2 - 4 \cdot 1 \cdot 3 \][/tex]

3. Simplify the expression:
[tex]\[ \Delta = 4 - 12 \][/tex]
[tex]\[ \Delta = -8 \][/tex]

4. Interpret the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two different real-number solutions.
- If [tex]\(\Delta = 0\)[/tex], the quadratic equation has one real-number solution.
- If [tex]\(\Delta < 0\)[/tex], the quadratic equation has two different imaginary-number solutions.

Since the discriminant [tex]\(\Delta = -8\)[/tex], which is less than zero, the quadratic equation:
[tex]\[ x^2 + 2x + 3 = 0 \][/tex]
has two different imaginary-number solutions.

Therefore, the discriminant is [tex]\(-8\)[/tex], and the quadratic equation has two different imaginary-number solutions.